In order that a relation $R$ defined on a non-empty set $A$ is an equivalence relation,it is sufficient,if $R$

Let $H$ be the set of all houses in a village where each house is faced in one of the directions,East,West,North,South. Let $R = \{ (x,y) | (x,y) \in H \times H \text{ and } x, y \text{ are faced in same direction} \}$. Then the relation $R$ is

Let $P$ be the relation defined on the set of all real numbers such that $P = \{(a,b) : \sec^2 a - \tan^2 b = 1\}$. Then $P$ is

Let $R$ be a relation on $\mathbb{R}$,given by $R = \{(a, b) : 3a - 3b + \sqrt{7} \text{ is an irrational number} \}$. Then $R$ is

Let $L$ be the set of all straight lines in the Euclidean plane. Two lines $l_1$ and $l_2$ are said to be related by the relation $R$ if and only if $l_1$ is parallel to $l_2$. Then the relation $R$ is

Show that the relation $R$ defined in the set $A$ of all triangles as $R = \{(T_{1}, T_{2}) : T_{1} \text{ is similar to } T_{2}\}$,is an equivalence relation. Consider three right-angled triangles $T_{1}$ with sides $3, 4, 5$,$T_{2}$ with sides $5, 12, 13$,and $T_{3}$ with sides $6, 8, 10$. Which triangles among $T_{1}, T_{2}$,and $T_{3}$ are related?